Homomorphic Encryption (HE)

Supports unlimited operations of a single type—either addition or multiplication, but not both. RSA encryption is a classic example, supporting multiplicative homomorphism. Paillier cryptosystem supports additive homomorphism and is widely used in federated learning for securely aggregating model weight updates. PHE schemes are computationally efficient and practical for specific, well-defined workflows like secure aggregation in cross-silo medical imaging consortia.

Supports both addition and multiplication but only for a limited number of operations before noise corrupts the ciphertext. SHE schemes are built on lattice-based cryptography and serve as the foundational stepping stone to fully homomorphic encryption. In diagnostic AI pipelines, SHE can handle bounded computations like evaluating a shallow neural network layer on encrypted patient scans.

Supports arbitrary computation on ciphertexts with unlimited additions and multiplications. FHE enables a hospital to send an encrypted CT scan to a cloud AI service, have a diagnostic model run inference, and receive an encrypted result—all without the cloud provider ever seeing the raw image. The primary bottleneck remains computational overhead, with operations on encrypted data being 10,000x to 1,000,000x slower than plaintext equivalents.

Modern HE schemes derive their security from the hardness of lattice problems, specifically the Learning With Errors (LWE) and Ring-LWE assumptions. These problems are believed to be resistant to attacks by both classical and quantum computers, making HE a post-quantum cryptographic primitive. This is critical for long-term protection of medical records that must remain confidential for decades.

Every HE operation adds a small amount of mathematical noise to the ciphertext. If noise exceeds a threshold, decryption fails. Bootstrapping, introduced by Gentry in 2009, is the breakthrough technique that evaluates the decryption circuit homomorphically to reset noise levels, enabling unlimited computation. This remains the most computationally expensive operation in FHE, often dominating runtime in complex medical image analysis workflows.

HE complements federated learning by protecting both model updates in transit and inference on deployed models. A practical architecture: multiple hospitals encrypt their local model gradients with HE, a central server performs homomorphic aggregation on the encrypted updates, and only the final encrypted global model is distributed. This provides defense-in-depth against gradient leakage attacks that could reconstruct patient MRI scans from plaintext model updates.

The most powerful class of HE schemes supporting arbitrary computation (unlimited additions and multiplications) on ciphertexts. FHE enables a server to execute any algorithm on encrypted data without ever decrypting it. Key characteristics:

• Supports unbounded depth circuits
• Based on lattice cryptography (e.g., Gentry's bootstrapping)
• Current schemes: CKKS, BGV, TFHE
• Performance trade-off: 4-6 orders of magnitude slower than plaintext computation
• Ideal for complex diagnostic model inference on encrypted patient scans

Schemes supporting only one operation type (addition OR multiplication) on ciphertexts. PHE is computationally lightweight and practical for specific use cases. Examples:

• RSA: Multiplicatively homomorphic
• Paillier: Additively homomorphic, widely used in secure aggregation
• ElGamal: Multiplicatively homomorphic
• Common in federated learning for securely summing model weight updates from multiple hospitals without revealing individual contributions

An intermediate scheme supporting limited-depth circuits with both addition and multiplication before noise overwhelms the ciphertext. SHE avoids the heavy computational cost of bootstrapping. Use case:

• Evaluating shallow neural network layers on encrypted medical images
• Computing statistical measures (mean, variance) on encrypted patient data
• Noise management: Each multiplication increases ciphertext noise exponentially
• Serves as the foundation for leveled FHE constructions like BGV and BFV

The critical technique invented by Craig Gentry in 2009 that enables FHE by refreshing ciphertext noise. Bootstrapping evaluates the decryption circuit homomorphically, resetting noise levels to permit unlimited computation. Performance impact:

• Single bootstrapping operation: milliseconds to seconds
• Major bottleneck in FHE throughput
• Recent advances (TFHE, FHEW) have dramatically reduced latency
• Enables deep neural network inference on fully encrypted diagnostic images